(Solved): Figure 2. Block Diagram of Cruise Control Closed-Loop System The transfer function for the vehicle ...

Figure 2. Block Diagram of Cruise Control Closed-Loop System The transfer function for the vehicle itself is unchanged from the previous part of this lab. We again will consider the car to have a mass of $m=1000\mathrm{kg}$, and the coefficient used for the resistance is $b=100\mathrm{N}?\sec /\mathrm{m}$. The system transfer function is therefore: $G_S=\frac{V(s)}{F(s)}=\frac{1}{1000s+100}$ The controller-actuator will now have both a proportional component and an integral component. This is represented by: $G_{CA}=\frac{F(s)}{E(s)}=K_P+\frac{K_I}{s}$ Again, $K_P$ is the constant the represents the proportional component, while $K_l$ is the constant that is multiplied by the integral of the error (represented by $1/5$ in a Laplace transform). Problem 1 (15 points): For the closed loop system with the vehicle and PI controller-actuator transfer functions given above, show that the block diagram reduces to: $V(s)=R(s)\left[\frac{K_{P}s+K_I}{1000s^2+\left(100+K_P\right)s+K_I}\right]$ Show the steps in your derivation in order to receive credit.